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Parseval's Relation


Let F(nu) and G(nu) be the Fourier transforms of f(t) and g(t), respectively. Then

 int_(-infty)^inftyf(t)g^_(t)dt 
=int_(-infty)^infty[int_(-infty)^inftyF(nu)e^(-2piinut)dnu][int_(-infty)^inftyG^_(nu^')e^(2piinu^'t)dnu^']dt 
=int_(-infty)^inftyF(nu)int_(-infty)^inftyG^_(nu^')[int_(-infty)^inftye^(2piit(nu^'-nu))dt]dnu^'dnu 
=int_(-infty)^inftyF(nu)[int_(-infty)^inftyG^_(nu^')delta(nu^'-nu)dnu^']dnu 
=int_(-infty)^inftyF(nu)G^_(nu)dnu,

where z^_ denotes the complex conjugate.


See also

Fourier Transform, Parseval's Theorem

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 425, 1985.

Referenced on Wolfram|Alpha

Parseval's Relation

Cite this as:

Weisstein, Eric W. "Parseval's Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParsevalsRelation.html

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